For \(q=1\)  and \(k=2\)  we have the sequence of Fibonacci numbers \((F_n)_{n\ge 0}\)  defined recurrencely by \(F_{n+1}=F_n+F_{n-1}\)  with initial conditions \(F_0=0\)  and \(F_1=1\) . Such sequence have been studied and generalized by many authors, see [1]}, [2]}, [3]}, [4]}. Some of these generalizations have received greater prominence over the years. Among these, it is worth mentioning here that the usually called sequences of \(k\) -generalized Fibonacci numbers or \(k\) -bonacci number occur here when \(q=1\)  and \(k\ge 2\) , and satisfy the recurrence relation
\(F_{n}^{(k)}=F_{n-1}^{(k)}+\cdots + F_{n-k}^{(k)}, \ \forall \ n\ge 2,\) 
